A local corrections algorithm for solving Poisson’s equation in three dimensions

Abstract

We present a second-order accurate algorithm for solving the free-space Poisson's equation on a locally-refined nested grid hierarchy in three dimensions. Our approach is based on linear superposition of local convolutions of localized charge distributions, with the nonlocal coupling represented on coarser grids. The representation of the nonlocal coupling on the local solutions is based on Anderson's Method of Local Corrections and does not require iteration between different resolutions. A distributed-memory parallel implementation of this method is observed to have a computational cost per grid point less than three times that of a standard FFT-based method on a uniform grid of the same resolution, and scales well up to 1024 processors.

A free-space adaptive FMM-Based PDE solver in three dimensions

“…Results are shown in Table 4 Example 3. For our third example, we replicate an experiment from [47] for a highly oscillatory force with discontinuities along multiple surfaces, setting…”

“…In order to compare our results to [47], we use the error metric introduced there. Let B be the vector of errors calculated as the difference between the calculated and exact solutions on B and calculate the following norm over all leaf boxes.…”

“…While the performance of our code cannot be compared easily to the optimized and parallelized scheme presented in [47], we have implemented schemes accurate to much higher order. Thus, as expected, we are able to reach comparable accuracies with significantly fewer points.…”

“…Compared to the state-of-the-art technique [47], our method is accurate to higher order and therefore solves similar problems with fewer degrees of freedom, and the work per point is approximately the same. Our current implementation uses OpenMP (but not MPI), although we expect the extension to be straightforward, as our solver is built on top of the MPI-based code of [61].…”

“…In [4], the method of local corrections (MLC) [3] was combined with multigrid methods to solve the Poisson equation on a hierarchy of nested grids. The fastest free-space Poisson solver for three-dimensional problems of which we are aware is described in [47]. It first solves local Poisson problems on fine grids using FFT-based techniques and then couples together the solutions on coarser grids using MLC.…”